We investigate extensions of dependence logic with generalized quantifiers. We also introduce and investigate the notion of a generalized atom. We define a system of semantics that can accommodate variants of dependence logic, possibly extended with generalized quantifiers and generalized atoms, under the same umbrella framework. The semantics is based on pairs of teams, or double teams. We also devise a game-theoretic semantics equivalent to the double team semantics. We make use of the double team semantics by defining a logic $$\hbox {DC}^2$$ DC 2 which canonically fuses together two-variable dependence logic $$\hbox {D}^2$$ D 2 and two-variable logic with counting quantifiers $$\hbox {FOC}^2$$ FOC 2 . We establish that the satisfiability and finite satisfiability problems of $$\hbox {DC}^2$$ DC 2 are complete for $$\hbox {NEXPTIME}$$ NEXPTIME
Keywords Team semantics  Dependence logic  Generalized quantifiers  Game-theoretic semantics
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DOI 10.1007/s10849-015-9217-4
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Philosophical Investigations.Ludwig Josef Johann Wittgenstein - 1953 - New York, NY, USA: Wiley-Blackwell.
Compositional Semantics for a Language of Imperfect Information.W. Hodges - 1997 - Logic Journal of the IGPL 5 (4):539-563.
Dependence and Independence.Erich Grädel & Jouko Väänänen - 2013 - Studia Logica 101 (2):399-410.

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Modal Independence Logic.Juha Kontinen, Julian-Steffen Müller, Henning Schnoor & Heribert Vollmer - 2014 - In Rajeev Goré, Barteld Kooi & Agi Kurucz (eds.), Advances in Modal Logic, Volume 10. CSLI Publications. pp. 353-372.
On Definability of Team Relations with K-Invariant Atoms.Raine Rönnholm - 2022 - Annals of Pure and Applied Logic 173 (10):103136.

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